How to Define "and"- and "or"-Operations for Intuitionistic and Picture Fuzzy Sets

The traditional fuzzy logic does not distinguish between the cases when we know nothing about a statement S and the cases when we have equally convincing arguments for S and for its negation ¬S: in both cases, we assign the degree 0.5 to such a statement S. This distinction is provided by intuitionistic fuzzy logic, when to describe our degree of confidence in a statement S, we use two numbers a+ and a− that characterize our degree of confidence in S and in ¬S. An even more detailed distinction is provided by picture fuzzy logic, in which we differentiate between cases when we are still trying to understand the truth value and cases when we are no longer interested. The question is how to extend “and”and “or”-operations to these more general logics. In this paper, we provide a general idea for such extension, an idea that explain several extensions that have been proposed and successfully used. 1 Formulation of the Problem Traditional fuzzy logic: brief reminder. In the traditional fuzzy logic (see, e.g., [2, 4, 5, 6, 7, 8]), we describe our degree of confidence in a statement A by a number a ∈ [0, 1], so that: • 0 means no confidence, • 1 means full confidence, and • intermediate values describe intermediate degrees of confidence. In practice, there is often a need to estimate the degree of confidence in composite statements like A&B and A ∨B. There are many such statements, so it is not feasible to ask the experts about all of them. Instead, we must